Polymeric membranes are ubiquitous in transport applications including gas separation, water desalination, solid-state batteries, and fuel cells. The transport mechanism in certain classes of membranes is well understood. It is generally believed that polymeric membranes used in gas separation follow the solution-diffusion model. However, the transport mechanism in other membranes such as the ones used in water desalination is less clear as both the solution-diffusion and hydrodynamic models have been proposed. In this contribution, we study the structure as well as the water and polymer dynamics of several polymeric membranes using small-angle and quasi-elastic neutron scattering. We demonstrate that on the scale of a few nanometers, water appears to diffuse through a desalination membrane at a rate comparable to bulk water, while water diffuses at a rate that is a factor of 2 slower in an anion-exchange membrane. These results illustrate the importance of studying both the structure and dynamics of membrane materials and have important implications in understanding the transport mechanisms that differentiate the different types of polymeric membranes.